Convex Semi-Regular Polytopes

Convex Semi-Regular Polytopes

DOI: 10.4018/978-1-5225-4108-0.ch004


The geometry of polytopes of higher dimension having deviations from the conditions for the correctness of the geometric figure is considered. These deviations reflect the shapes of the molecules of the chemical compounds studied in Chapters 1-3. From the validity conditions in all cases the condition of topological equivalence of the vertices of the polytope is preserved. All these polytopes are called semi-regular. We study the hierarchical filling of spaces with polytopes of higher dimension, different from the well-known filling of spaces with spheres of constant diameter. The considered fillings characterize the distribution of atoms in nanostructures, in which the growth centers are distributed throughout the volume of the structure.
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Golden Hyper-Rhombohedron: Translational Basis Of Quasi-Crystals

Finding in 1982 of ordered structures deprived (as it seemed) of translational symmetry (Shechtman et al., 1984), next called “quasicrystals”, had marked the beginning of numerous cycles of papers and books devoted to the experimental and theoretical study of these unusual materials. But later it was found that the diffraction patterns of quasicrystals have a latent periodicity (Zhizhin, 2014), if we consider the diffraction pattern as a projection of a structure from a space of higher dimension. Figure 1 shows a typical diffraction pattern of intermetallic compounds.

Figure 1.

Electron diffraction pattern of compound Al72Ni20Co8 (Eiji Abe et al., 2004)


Similar structures have other intermetallic compounds involving transition metals, for example: Al6Mn (Shechtman et al., 1984), Al70Fe20W10 (Mukhopadhyay et al., 1993), Ti54Zr26Ni20 (Zhang & Kelton, 1993).

Figure 1 shows that the luminous points, which are a reflection of light from the impact of the electron beam, form five families of parallel lines oriented with respect to each other at angles of a multiple of 72 degrees. The distances between the parallel lines and the angles are determined by the golden section. A geometrical model of the structure of the diffraction patterns of quasicrystals was constructed (Shevchenko, Zhizhin & Mackey, 2013a; Shevchenko, Zhizhin & Mackey, 2013b; Zhizhin, 2014; Zhizhin & Diudea, 2016). It was shown that the elementary cell of this geometric structure is a polytope of dimension 4, which was called a gold hyper-rhombohedron. This cell is plotted on the diffraction pattern in Figure 1 with solid segments of light lines. I can see that it passes through the luminous points of the diffraction pattern observing its geometry. This cell fills the entire space with the translation reflected by the diffraction pattern. To determine the regularities of this cell, it is depicted in Figure 2 on an enlarged scale.

Figure 2.

Golden hyper-rhombohedron


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